Bayesian spectral likelihood for hydrological parameter inference

Abstract

This paper proposes a spectral domain likelihood function for the Bayesian estimation of hydrological model parameters from a time series of model residuals. The spectral domain error model is based on the power-density spectrum (PDS) of the stochastic process assumed to describe residual errors. The Bayesian spectral domain likelihood (BSL) is mathematically equivalent to the corresponding Bayesian time domain likelihood (BTL) and yields the same inference when all residual error assumptions are satisfied (and all residual error parameters are inferred). However, the BSL likelihood function does not depend on the residual error distribution in the original time domain, which offers a theoretical advantage in terms of robustness for hydrological parameter inference. The theoretical properties of BSL are demonstrated and compared to BTL and a previously proposed spectral likelihood by Montanari and Toth (2007), using a set of synthetic case studies and a real case study based on the Leaf River catchment in the U.S. The empirical analyses confirm the theoretical properties of BSL when applied to heteroscedastic and autocorrelated error models (where heteroscedasticity is represented using the log-transformation and autocorrelation is represented using an AR(1) process). Unlike MTL, the use of BSL did not introduce additional parametric uncertainty compared to BTL. Future work will explore the application of BSL to challenging modeling scenarios in arid catchments and “indirect” calibration with nonconcomitant input/output time series.